Multiplicity results for a fourth-order semilinear elliptic problem. (English) Zbl 0898.35032

The paper studies the fourth order elliptic problem \[ \Delta^2 u+c\Delta u= bg (x,u)\quad\text{in } \Omega, \quad\Delta u=0,\;u=0\quad \text{on } \partial \Omega \] by studying the critical points of the related functional \[ f_b(u)= {1\over 2} \left(\int_\Omega (\Delta u)^2-c \int_\Omega |\nabla u|^2\right) -b\int_\Omega G(x,u), \quad u\in H^2 (\Omega) \cap H^1_0 (\Omega) \] where \(G\) is an antiderivative of \(g\) with respect to \(u\). The existence of multiple critical points of \(f_b\) is established by applying the “linking” theorems of M. Schechter and K. Tintarev [Bull. Soc. Math. Belg., Ser. B 44, 249-261 (1992; Zbl 0785.58017)] and A. Marino, A. M. Micheletti and A. Pistoia [Topol. Methods Nonlinear Anal. 4, 289-339 (1994; Zbl 0844.35035)]. The paper finishes with an application of Leray-Schauder degree to prove (in the nondegenerate case) the existence of at least three nontrivial solutions of the equation in the special case \(g=\max \{(u+1)^+ -1,0\}\).


35J35 Variational methods for higher-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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